If you want more than a token bit of XBLA and DLC then you are going to need to go down the JTAG/RGH route. JTAG and RGH have more or less identical end results although JTAG can be done with the results of a trip to a basic electronics shop where RGH requires some more fancy soldering and you to buy in some hardware. JTAG requires a machine that has not been updated since August 11th 2009 where RGH can be done on more recent versions of the dashboard (although there is a cutoff for later versions unless you otherwise have keys and NAND access which you would not have unless you have otherwise done the hacks). Chances are you will not find a JTAG capable machine these days* and RGH machines, assuming another version of the hack does not come along, are also getting a bit harder to come by (14699 or last December's update- I am not sure when new games started requiring it but it is usually a couple of months later, Borderlands 2 which only just hit requires the next version up though).
JTAG definitely stops live and RGH can not be used with live but there are ways to do dual NAND and have a version that goes on live, dual nand is a pain to do and keep current so I do not suggest it for most people.
*you might see them offered for sale, I personally suggest nothing that is not a Jasper motherboard (the seller will know and if they do not know then walk away) be purchased for fear of RRoD but others might take a Falcon motherboard.
Most people opt to modify the DVD drive though (which only does copied DVDs and the DLC installation mentioned in a moment), these days that comes in two types.
1) Traditional DVD modding, here you find the DVD keys (different to the keys mentioned a moment ago) and use them to build a custom firmware for the DVD drive. Tutorials
http://jungleflasher.../downloads.html Generally the older the system (my stance on motherboards does not change here) the easier the mod unless it has been updated past July 2011's update or it is a 360 slims (some are not so bad but they require thought where most earlier drives could be done without really thinking about it).
You can buy a standalone modding tool to do it (reasonably expensive) or you can sort appropriate hardware out (less expensive and stuff you can buy from mainstream retailers). These days most big releases (pretty much every big game since August last year) will also require you to have a specially modded DVD drive for burning, they are not that expensive, not that rare (although they are getting ever so slightly harder to get your hands on as the months drag on) and can still be used as normal DVD drives at other points. All of it requires dual layer discs to work.
2) Optical disc drive emulation (ODDE). If you ever used a WODE for the Wii it is pretty much that but for the 360 instead. Some have claimed they are not as safe for xbox live but if you do not care about this then it is all good. Here you stick your keys into the ODDE and then hook a USB drive up to it.
The token XBLA and DLC I mentioned is stuff like the borderlands double pack, fallout expansions and such which came as disc installers and can be stuck on the hard drive via other means (this is as close to a softmod as you can get for the 360). This is perfectly safe and can even be done on stock machines. There is not a lot else these days worth doing (profile hacking is fairly pointless as it gains nothing), before there was stuff like profile sharing DLC and XBLA but MS put a stop to that.
